A Simple Homotopy Proximal Mapping for Compressive Sensing

TL;DR

This paper introduces a simple homotopy proximal mapping algorithm for compressive sensing that guarantees global linear convergence under various measurement conditions, outperforming previous methods in efficiency and accuracy.

Contribution

The paper proposes a novel homotopy proximal mapping algorithm with proven linear convergence for compressive sensing, improving theoretical bounds and practical efficiency.

Findings

Algorithm achieves linear convergence under RIP conditions.

Outperforms previous methods in accuracy and speed.

Empirical results support theoretical claims.

Abstract

In this paper, we present a novel yet simple homotopy proximal mapping algorithm for compressive sensing. The algorithm adopts a simple proximal mapping of the $\ell_1$ norm at each iteration and gradually reduces the regularization parameter for the $\ell_1$ norm. We prove a global linear convergence of the proposed homotopy proximal mapping (HPM) algorithm for solving compressive sensing under three different settings (i) sparse signal recovery under noiseless measurements, (ii) sparse signal recovery under noisy measurements, and (iii) nearly-sparse signal recovery under sub-gaussian noisy measurements. In particular, we show that when the measurement matrix satisfies Restricted Isometric Properties (RIP), our theoretical results in settings (i) and (ii) almost recover the best condition on the RIP constants for compressive sensing. In addition, in setting (iii), our results for sparse signal recovery are better than the previous results, and furthermore our analysis explicitly exhibits that more observations lead to not only more accurate recovery but also faster convergence. Compared with previous studies on linear convergence for sparse signal recovery, our algorithm is simple and efficient, and our results are better and provide more insights. Finally our empirical studies provide further support for the proposed homotopy proximal mapping algorithm and verify the theoretical results.